Abstract : We present and axiomatize several update rules for probabilities (and preferences) where there is no unique additive prior. In the context of non-additive probabilities we define and axiomatize Bayesian update rules; in the context of multiple (additive) priors we define maximum likelihood rules. It turns out that for decision makers which can be described by both theories, the two approaches coincide. Thus, we suggest a pseudo-Bastion foundation to classical statistics, which may also motivate alternative statistical inference techniques, and provide an axiomatically-based ambiguous belies update rule, which is needed for their application in many economic theory models.